The three Weyl components form one fundamental representation of the global $SU(3)$, so coupling $G$ to background gauge fields reveals a nonzero $SU(3)^3$ <'t Hooft anomaly>. This does not destroy the global symmetry in flat space or make the theory inconsistent; it constrains any infrared description through <'t Hooft anomaly matching>. If $G$ were instead intended as a dynamical gauge symmetry, the same uncancelled <gauge anomaly> would make the model inconsistent and additional chiral matter would be required. Ordinary quantum corrections cannot generate the forbidden superpotential, and the <non-renormalization theorem> protects its absence.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 307 / 1 / vii / Solution

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The supersymmetric non-renormalization theorem says that the Wilsonian superpotential receives no perturbative corrections. Holomorphy and spurion symmetries can often exclude nonperturbative corrections as well.


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 Incoming links: Non-renormalization theorem